Diagonal Matrix
A diagonal matrix is a special type of square matrix where all the non-diagonal elements are zero. That is, a matrix \( D \) is called a diagonal matrix if:
\[ D = \begin{bmatrix} d_{11} & 0 & 0 & \dots & 0 \\ 0 & d_{22} & 0 & \dots & 0 \\ 0 & 0 & d_{33} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & d_{nn} \end{bmatrix} \]
where \( d_{ii} \) (for \( i = 1, 2, \dots, n \)) are the diagonal elements, and all off-diagonal elements are zero.
Examples of Diagonal Matrices
1. A 2×2 diagonal matrix:
\[ D = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix} \]
2. A 3×3 diagonal matrix:
\[ D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]
3. A 4×4 diagonal matrix:
\[ D = \begin{bmatrix} 4 & 0 & 0 & 0 \\ 0 & 9 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \]
Properties of Diagonal Matrices
Property 1: Addition of Two Diagonal Matrices
The sum of two diagonal matrices of the same size results in another diagonal matrix.
Example: Consider two 3×3 diagonal matrices:
\[ D_1 = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -3 \end{bmatrix}, \quad D_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 6 \end{bmatrix} \]
Their sum is:
\[ D_1 + D_2 = \begin{bmatrix} 2+1 & 0 & 0 \\ 0 & 5+(-4) & 0 \\ 0 & 0 & -3+6 \end{bmatrix} = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]
Property 2: Multiplication of Two Diagonal Matrices
The product of two diagonal matrices of the same size results in another diagonal matrix where each diagonal element is the product of the corresponding diagonal elements.
Example: Using the same matrices \( D_1 \) and \( D_2 \):
\[ D_1 \times D_2 = \begin{bmatrix} 2 \times 1 & 0 & 0 \\ 0 & 5 \times (-4) & 0 \\ 0 & 0 & (-3) \times 6 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -20 & 0 \\ 0 & 0 & -18 \end{bmatrix} \]
Property 3: Inverse of a Diagonal Matrix
If a diagonal matrix \( D \) is invertible (i.e., none of its diagonal elements are zero), then its inverse is also a diagonal matrix with each diagonal element replaced by its reciprocal.
Example: Consider:
\[ D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]
The inverse is:
\[ D^{-1} = \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1/3 & 0 \\ 0 & 0 & 1/5 \end{bmatrix} \]
Property 4: Determinant of a Diagonal Matrix
The determinant of a diagonal matrix is the product of its diagonal elements.
Example: For:
\[ D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 7 \end{bmatrix} \]
The determinant is:
\[ \det(D) = 4 \times 2 \times 7 = 56 \]
Property 5: Diagonal Matrices in Eigenvalues and Eigenvectors
The eigenvalues of a diagonal matrix are simply the diagonal elements themselves, and the eigenvectors are the standard basis vectors.
Example: For:
\[ D = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]
The eigenvalues are \( 3, -1, 5 \), and the corresponding eigenvectors are:
\[ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \]